1549301742-The_Theory_of_Difference_Schemes__Samarskii

Heat conduction equation with constant coefficients 305

yields

(12)

• 
  
  
  
  • h^2 2 4 2
    y0 = ( L u - ll + <p) + ( O" - 0. 5) T L 1l + 12 L fl + 0 ( h + T ) ,
  
  
  
  

whence it follows that ·tf; = (u - 0.5) TL u + O(h^2 + T^2 ) for <p = .f

.f(x, ij+i; 2 ), since it= Lu+ .f. By virtue of the relations L 1£ = L^2 u+L.f =

u (^4 ) + .f" and L^2 u =Lit - L .f and expression (12) we are led to

By equating the expression in the square brackets to zero we find that

(14)

1 h^2

O"=----=O".

2 12 T *

For u = u * and <p = .f + / 2 h^2 L J scheme (II) generates an approximation

of O(h^4 + T^2 ), that is, y0 = O(h^4 + T^2 ). B~t the order of approximation of

this scheme remains unchanged if .f" will be replaced by lex = A .f, giving

<p = J + 112 ( h 2 A/) or

(15) <pi j -_ 6. ~ rj+112 i + ~ 12 (.t·j+112 i-1 +. f.j+112) i+1.

Observe that the last formula is rnuch n1ore sirnpler for later use.
Let C;;' ( D) be the class of functions with n1 derivatives in x and n
derivatives in t, all derivatives being continuous in D. Formulae ( 13 )-(14)
justify that scheme (II) provides approximations of

• O(h^2 + T^2 ) for u = 0.5, <p = J or <p = J + O(h^2 + T^2 ) ifu E Cj,


• O(h^2 + T) for any u #- 0.5, <p = J + O(h^2 + T), for instance, <p = J
  or <p = f if u E C'.L


• O(h^4 + T^2 ) for u = u. and <p specified by formula (15) if u E CJ.
  Scheme (II) with u. = u * and <p = J + / 2 h^2 A J is usually termed a higher-
  accuracy scheme. The requirement of the retention of the approximation
  order for a given value u guides a proper choice of the right-hand side <p.

Thus, for u = 0.5 it is taken to be <p = 0.5 (} + .f), <p = .f and more.
It is easily seen from (13) that the error O(h^2 + T^2 ) may be also
achieved for u #- 0.5 if we keep u = 0.5 + h^2 a/T, where CY is an arbitrary
constant independent of h and T. Certainly, in this case u depends on h
and T. The arbitrariness in the choice of CY is limited by the condition of
stability of the scheme (a result is ensured under the constraint CY > -~,
for more detail see Section 4).